Paper · Sep 26, 2025

Noncollapsing for Curvature Flows with Inhomogeneous Speeds

Weimin Sheng, Ye Zhu

math.DGWider flowsarXiv:2509.21830
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Abstract

We study closed, embedded hypersurfaces in Euclidean space evolving by fully nonlinear curvature flows, whose speed is given by a symmetric, monotone increasing, -homogeneous, positive underlying speed function composed with a modulating function . Under the assumption that is convex or inverse-concave and that satisfies the corresponding structural conditions, we establish exterior noncollapsing estimates for the flow. The main difficulty stems from the nonlinearity of the evolution equation satisfied in the viscosity sense by the exscribed curvature, whereas in previous works it is a solution to the linearized flow. Moreover, in the case where is inverse-concave, we refine Andrews and Langford's argument for the interior case.

Comments: 19 pages

Other curvature flows(Non)collapsingHomogeneous spaces

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    Weimin Sheng and Ye ZhuWider flows PDF

    Topics (Non)collapsing, Other curvature flows · Same authors Weimin Sheng, Ye Zhu · Shared terms “exterior noncollapsing”, “nonlinear curvature”, “fully nonlinear”

  2. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Same author Weimin Sheng · Shared terms “nonlinear curvature”, “fully nonlinear”

  3. Non-homogeneous curvature flows in hyperbolic space

    Hongyi Sheng, Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Same author Weimin Sheng · Shared terms “structural condition”

  4. Non-homogeneous curvature flows in a hemisphere

    Hongyi Sheng, Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Same author Weimin Sheng · Shared terms “structural condition”

  5. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows PDF

    Topics (Non)collapsing, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “noncollapsing”

  6. Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

    Mat Langford and James McCoyWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “speed function”

  7. Collapsed manifolds with local Ricci bounded covering geometry

    Xiaochun Rong PDF

    Topics (Non)collapsing, Homogeneous spaces

  8. A flow approach to the Musielak-Orlicz-Gauss image problem

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.